L11a516

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_16,5,17,6 X_18,9,19,10 X_10,17,11,18 X_20,15,21,16 X_14,4,15,3 X_4,22,5,21 X₂₇₃₈ X_22,12,13,11 X_12,14,7,13 X_6,19,1,20
Gauss code	{1, -8, 6, -7, 2, -11}, {8, -1, 3, -4, 9, -10}, {10, -6, 5, -2, 4, -3, 11, -5, 7, -9}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)(t(3)-1)\left(t(3)^{2}t(2)^{2}+t(1)t(3)t(2)^{2}-2t(3)t(2)^{2}+t(2)^{2}+t(1)t(3)^{2}t(2)-t(3)^{2}t(2)+t(1)t(2)-2t(1)t(3)t(2)+2t(3)t(2)-t(2)-t(1)t(3)^{2}-t(1)+2t(1)t(3)-t(3)\right)}{t(1)t(2)t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{4}-4q^{3}+9q^{2}-15q+21-22q^{-1}+24q^{-2}-19q^{-3}+15q^{-4}-9q^{-5}+4q^{-6}-q^{-7}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-z^{2}a^{6}-a^{6}+3z^{4}a^{4}+6z^{2}a^{4}+a^{4}z^{-2}+3a^{4}-2z^{6}a^{2}-6z^{4}a^{2}-7z^{2}a^{2}-2a^{2}z^{-2}-4a^{2}-z^{6}-z^{4}+z^{2}+z^{-2}+2+z^{4}a^{-2}+z^{2}a^{-2}$ (db)
Kauffman polynomial	$a^{7}z^{7}-3a^{7}z^{5}+3a^{7}z^{3}-a^{7}z+4a^{6}z^{8}-13a^{6}z^{6}+14a^{6}z^{4}-7a^{6}z^{2}+2a^{6}+6a^{5}z^{9}-16a^{5}z^{7}+7a^{5}z^{5}+6a^{5}z^{3}-3a^{5}z+3a^{4}z^{10}+7a^{4}z^{8}-48a^{4}z^{6}+60a^{4}z^{4}+z^{4}a^{-4}-29a^{4}z^{2}-a^{4}z^{-2}+7a^{4}+17a^{3}z^{9}-44a^{3}z^{7}+24a^{3}z^{5}+4z^{5}a^{-3}+3a^{3}z^{3}-z^{3}a^{-3}-4a^{3}z+2a^{3}z^{-1}+3a^{2}z^{10}+19a^{2}z^{8}-73a^{2}z^{6}+9z^{6}a^{-2}+74a^{2}z^{4}-7z^{4}a^{-2}-33a^{2}z^{2}+2z^{2}a^{-2}-2a^{2}z^{-2}+7a^{2}+11az^{9}-13az^{7}+14z^{7}a^{-1}-8az^{5}-18z^{5}a^{-1}+8az^{3}+7z^{3}a^{-1}-2az+2az^{-1}+16z^{8}-29z^{6}+20z^{4}-9z^{2}-z^{-2}+3$ (db)

Khovanov Homology

The coefficients of the monomials

t^{r}q^{j}

are shown, along with their alternating sums

\chi

(fixed

j

, alternation over

r

).

\		r
	\
j		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

9

1

7

3

-3

5

6

1

5

3

9

3

-6

1

12

6

-1

12

11

-1

-3

12

10

2

-5

9

14

5

-7

6

10

-4

-9

3

9

6

-11

1

6

-5

-13

3

-15

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-1$	$i=1$
$r=-7$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-4$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-3$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=-2$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{12}$
$r=-1$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=0$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{12}$
$r=1$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=2$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$